Effective Dissipation and Turbulence in Spectrally Truncated Euler Flows

Abstract

A new transient regime in the relaxation towards absolute equilibrium of the conservative and time-reversible 3-D Euler equation with high-wavenumber spectral truncation is characterized. Large-scale dissipative effects, caused by the thermalized modes that spontaneously appear between a transition wavenumber and the maximum wavenumber, are calculated using fluctuation dissipation relations. The large-scale dynamics is found to be similar to that of high-Reynolds number Navier-Stokes equations and thus to obey (at least approximately) Kolmogorov scaling.PACS numbers: 47.27. Eq,05.20.Jj, 83.60.Df Turbulence has been observed in inviscid and conservative systems, in the context of (compressible) lowtemperature superfluid turbulence [1,2,3]. This behavior has also been reproduced using simple (incompressible) Biot-Savart vortex methods, which amount to Eulerian dynamics with ad hoc vortex reconnection [4]. The purpose of the present letter is to study the dynamics of spectrally truncated 3-D incompressible Euler flows. Our main result is that the inviscid and conservative Euler equation, with a high-wavenumber spectral truncation, has long-lasting transients which behave just as those of the dissipative (with generalized dissipation) Navier-Stokes equation. This is so because the thermalized modes between some transition wavenumber and the maximum wavenumber can act as a fictitious microworld providing an effective viscosity to the modes with wavenumbers below the transition wavenumber.We thus study general solutions to the finite system of ordinary differential equations for the complex variableŝwhereThis system is time-reversible and exactly conserves the kinetic energy E = k E(k, t), where the energy spectrum E(k, t) is defined by averagingv(k ′ , t) on spherical shells of width ∆k = 1,The discrete equations (1) are classically obtained [5] by performing a Galerkin truncation (v(k) = 0 for sup α |k α | ≤ k max ) on the Fourier transform v(x, t) = v(k, t)e ik·x of a spatially periodic velocity field obeying the (unit density) three-dimensional incompressible Euler equations,The short-time, spectrally-converged truncated Eulerian dynamics (1) has been studied [6,7] to obtain numerical evidence for or against blowup of the original (untruncated) Euler equations (3). We will study here the behavior of solutions of (1) when spectral convergence to solutions of (3) is lost. Long-time truncated Eulerian dynamics is relevant to the limitations of standard simulations of high Reynolds number (small viscosity) turbulence which are performed using Galerkin truncations of the Navier-Stokes equation [8].Equations (1) are solved numerically using standard [9] pseudo-spectral methods with resolution N . The solutions are dealiased by spectrally truncating the modes for which at least one wave-vector component exceeds N/3 (thus a 1600 3 run is truncated at k max = 534). This method allows the exact evaluation of the Galerkin convolution in (1) in only N 3 log N operations. Time marching is done with a second-order ...

Two-fluid model of the truncated Euler equations

“…These quantities are shown in figure 2, where that the mean heat is seen to increases in time, due to the energy coming from the large eddies, as was shown precedently in [2]. The relative fluctuation ∆Q/Q is seen to decrease from 0.9 to 0.2.…”

“…Recently, Cichowlas et al [2,3] observed that the Euler equation, with a very large (several hundreds) spectral truncation wavenumber k max , has longlasting transients which behave just as those of high-Reynolds number viscous flow; in particular they found an approximately k −5/3 inertial range followed by a dissipative range. How is such a behavior possible?…”

The quark-gluon plasma and the stochastic interpretation of quantum mechanics

“…Three dimensional simulations of truncated, incompressible, and inviscid Euler equations have shown similar chaotic turbulent results [24], including an interesting transient Kolmogorov scaling regime resembling a viscous fluid caused by the flow of energy from large scales to small as required by ergodic behavior. This result shows that some aspects of viscosity can emerge from an inviscid fluid theory in a chaotic turbulent regime.…”

“…We note that turbulent inviscid fluids have exhibited emergent viscous behavior in numerical simulation [24], and so the nonzero value for / s η as measured at RHIC, already the smallest viscosity ever measured for any material, does not necessarily rule out the possibility of an even lower viscosity or even zero viscosity in some protofluid whose highly developed turbulent state is being observed at RHIC. We are interested here in exploring the possibility that turbulence in such a protofluid might lead to quantum behavior as an emergent and purely classical phenomenon.…”

Gravitational instability via the Schrödinger equation

“…A related but converse phenomenon arises in Eulerian fluids (i.e. fluids without intrinsic viscosity) when they are described in terms of a truncated spectrum (Cichowlas et al 2005).…”